VOLUME 76, NUMBER 3 PHYSICAL REVIEW LETTERS 15 JANUARY 1996 Charge Density Wave in Two-Dimensional Electron Liquid in Weak Magnetic Field A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii Theoretical Physics Institute, University of Minnesota, 116 Church Street Southeast, Minneapolis, Minnesota 55455 (Received 7 August 1995) We study the ground state of a clean two-dimensional electron liquid in a weak magnetic field where N ¿ 1 lower Landau levels are completely filled and the upper level is partially filled. It is shown that the electrons at the upper Landau level form domains with filling factors equal to 1 and zero. The domains alternate with a spatial period of order of the cyclotron radius, which is much larger than the interparticle distance at the upper Landau level. The one-particle density of states, which can be probed by tunneling experiments, is shown to have a gap linearly dependent on the magnetic field in the limit of large N. PACS numbers: 73.20.Dx, 73.40.Gk, 73.40.Hm The nature of the ground state of an interacting twodimensional (2D) electron gas in a magnetic field has attracted much attention. The studies have been focused mostly on the case of very strong magnetic fields where only the lowest Landau level (LL) is occupied, so that the filling factor n ­ kF2 l 2 does not exceed unity (here kF is the Fermi wave vector of the 2D gas in zero magnetic field and l is the magnetic length, l 2 ­ h̄ymvc ). The physics at the lowest LL turned out to be so rich that, perhaps, only at n ­ 1 does the ground state have a simple structure; namely, it corresponds to one fully occupied spin subband of the lowest LL. The charge density in such a state is uniform. The case of a partial filling, n , 1, is much more interesting. Using the Hartree-Fock (HF) approximation, Fukuyama, Platzman, and Anderson  found that a uniform uncorrelated spin-polarized electron liquid (UEL) is unstable against the formation of a charge density wave (CDW) at wave vectors larger than 0.79l 21 . The optimal CDW period was later found to coincide with that of the classical Wigner crystal (WC) . Subsequently, however, it turned out that non-HF trial states suggested by Laughlin  for n ­ 1y3 and 1y5 to explain the fractional quantum Hall effect are lower in energy by a few percent. The Laughlin states were further interpreted in terms of an integer number of fully occupied LL’s of new quasiparticles, composite fermions . This concept was then applied to even denominator fractions . Thus, although the HF approximation gives a rather accurate estimate of the energy, it fails to describe important correlations in a partially filled lowest LL. Recently, the requirement of complete spin polarization in the ground state was also reconsidered. It was found that a partially filled lowest LL may contain Skyrmions . In this Letter we consider the case of weak magnetic fields or high LL numbers N. There is growing evidence from analytical and numerical calculations that fractional states, composite fermions, and Skyrmions are restricted to the lowest and the first excited LL’s (N ­ 0, 1) only (see Refs. [7–9]). We will present an additional argument in favor of this conclusion. This point of view is also 0031-9007y96y76(3)y499(4)$06.00 consistent with the experiment because none of those structures has been observed for N . 1. Before we proceed to the main subject of the paper, a partially filled upper LL, note that we can use the concept of LL’s only if the electron-electron interactions do not destroy the Landau quantization. For weak magnetic fields where the cyclotron gap h̄vc is small, this is far from being evident. To see that the LL mixing is indeed small one has to calculate the interaction energy per particle at the upper LL and verify that its absolute value is much smaller than h̄vc . The largest value of the interaction energy is attained at n ­ 2N 1 1 where the electron density at the upper LL is the largest. The interaction energy per particle is equal 1 to 2 2 Eex , where Eex is the exchange-enhanced gap for the spin-flip excitations  at n ­ 2N 1 1 (it determines, e.g., the activation energy between spin-resolved quantum Hall resistivity peaks). Aleiner and Glazman (AG)  calculated Eex to be µ p ∂ rs h̄vc 2 2 rs ø 1 , (1) 1 Eh , ln Eex ­ p rs 2p where Eh is the “hydrodynamic” term (see Ref. ) given by  Eh ­ h̄vc lnsNrs d . 2N 1 1 (2) The parameter rs entering these formulas is defined by p rs ­ 2ykF aB , aB ­ h̄ 2 kyme2 being the effective Bohr radius. In realistic samples rs , 1 but even at such rs the ratio Eex yh̄vc is still rather small. Therefore even at weak magnetic fields the cyclotron motion is preserved and the mixing of the LL’s is small. Note that the first term in Eex linearly depends on the magnetic field, whereas Eh has an approximately quadratic dependence. Since we chose to rely on the HF approximation, a natural turn of thought is to consider a WC-type state whose wave function is given by [9,13] Y y cRi j0N l , (3) jCl ­ i © 1996 The American Physical Society 499 VOLUME 76, NUMBER 3 PHYSICAL REVIEW LETTERS 15 JANUARY 1996 y where j0N l stands for N completely filled LL’s and cR is the creation operator for a certain one-particle state, called a coherent state . The modulus of the coherent state wave function is not small only within a distance l off the classical cyclotron orbit with the center at the point R and radius Rc ­ kF l 2 . In the HF WC state Ri coincide with the sites of a triangular lattice with density nN y2pl 2 , where nN ; n 2 2N. From now on we con1 sider only nN # 2 , which suffices because of the electronhole symmetry. When nN is small, nN ø 1yN, the cyclotron orbits at neighboring lattice sites do not overlap, and the concept of the WC is natural. However, this concept was applied for overlapping orbits as well. According to AG, at N ¿ rs22 ¿ 1 and not too small nN , nN ¿ 1yNrs2 , the cohesive energy of the WC; i.e., the energy per particle at the upper LL with respect to that in the UEL of the same density, is given by  ∏ ∑p h̄vc 3 1 2 nN 2 WC 1 lnsNnN d 2 Eh . Ecoh ­ 2 16pN rs 2p 2 (4) Assuming that the WC is the ground state, AG found that the one-particle density of states (DOS) consists of two narrow peaks separated by the gap Eg ­ Eh (see also WC j are Ref. ). In the limit of large N, both Eg and jEcoh much smaller than Eex , and so AG concluded that there are two different scales for spin and charge excitations. In this Letter we claim that for nN ¿ 1yNrs2 the ground state is not the WC, but another type of CDW whose period is of order Rc . In contrast to the lowest LL, the optimal CDW period is much larger than the average distance between the electrons at the upper LL. The cohesive energy of the CDW has the scale Eex and is given by µ ∂ 0.3 1 2 nN CDW ø 2fsnN drs h̄vc ln 1 1 2 Eh , Ecoh rs 2 (5) 1 where fsnN d ø 0.03 at nN ­ 2 and fsnN d ~ nN at 1 1yNrs2 ø nN ø 2 . The DOS consists of two peaks (Van Hove singularities) at the edges of the spectrum, the 1 distance between them for nN , 2 being equal to ∂ µ rs h̄vc 0.3 p (6) 1 Eh . ln 1 1 Eg ø rs 2p Hence, we claim that all the important properties of the Nth LL are determined by the single scale Eex . WC CDW Let us compare Ecoh and Ecoh . The “hydrodynamic” term is the same in both. Hence one has to compare only the remaining terms. It is easy to see that the CDW state wins over the WC provided nN * 1yNrs2 . Our CDW state can be roughly approximated by a state (3), with Ri forming patterns shown in Fig. 1. The aggregation of many particles in large domains of size Rc allows the system to achieve a lower value of the exchange energy. At the same time, due to the fact that the domain 500 FIG. 1. CDW patterns. (a) Stripe pattern. (b) Bubble pattern. (c) WC. One cyclotron orbit is shown. separation is chosen according to the special ringlike shape of the wave functions at the upper LL, the actual charge density variations are not too large (of order 20%). Hence, the increase in the Hartree energy due to the domain formation is small. According to our numerical simulations for N ­ 5 and rs ­ 0.5, at nN . 0.3 the optimal CDW has a “stripe” structure [Fig. 1(a)]. At nN , 0.3 a “bubble” pattern [Fig. 1(b)] wins. The distance between the “bubbles” in this pattern is of order Rc and remains approximately the same as nN decreases. Correspondingly, p their diameter is given by ,Rc nN . At nN , 1yN where it becomes of order l the “bubbles” consist of single electrons, i.e., the CDW state becomes indistinguishable from the WC [Fig. 1(c)]. With further decrease in nN , the distance between the electrons increases. At this point we would like to address the issue of the fractional states at high LL’s. We believe that at nN ¿ 1yN the fractional states cannot compete with the CDW state. Indeed, the CDW state has a very low energy because of the correlations in the positions of the guiding centers on the length scale Rc , which is the largest length scale in a not too dilute system. In the fractional states, just like in the WC, these correlations have the length scale l. Based on the example of the WC, it seems very plausible that the correlations of this type are much less effective. On the other hand, there is no doubt that at nN ø 1yN the WC is the ground state. This leaves only a narrow window in the vicinity of nN ­ 1yN, where the fractional states may or may not appear. The novel ground state enables us to explain two interesting experimental findings. One is the magnitude of a pseudogap in the tunneling DOS, first observed in experiments on a single quantum well  and, recently, on double quantum well high-mobility GaAs systems [17,18]. The pseudogap Etun appears to be linear in magnetic field for 1 # N # 4 . Theoretically, the pseudogap is given by Etun ­ 2Eg . The additional factor of 2 arises because the tunneling DOS is the convolution of the DOS of the two wells. For the parameters of Ref.  Eq. (6) leads to Etun ø 0.52h̄vc , which compares favorably with the experimental value of 0.45h̄vc . In the experimental range of parameters the “hydrodynamic” term dominates, and our result is only 35% larger than that of AG, 2Eh . However, in the limit N ¿ 1 we predict a much wider pseudogap with a linear instead of an approximately quadratic dependence on the magnetic field. Note that even for 1 # N # 4 the dependence, which we predict, is not much different from the linear one. Recently, Levitov and Shytov  obtained an expression for Etun similar VOLUME 76, NUMBER 3 PHYSICAL REVIEW LETTERS but not identical to ours without studying the ground state of the system. We believe that only the CDW ground state can justify this type of expression. Another important application of the proposed picture concerns the conductivity peak width of the integer quantum Hall effect in high-mobility structures where the disorder is believed to be long range. A semiclassical electrostatic model of Efros  predicts that the electron liquid is compressible in a large fraction of the sample area. If the compressible liquid is considered to be metallic, then the conductivity peaks are necessarily wide , which is indeed observed at relatively high temperatures . However, it is well known that at low temperatures the peaks are narrow (see, e.g., Ref. ), which may result from the pinning of the compressible liquid . The fine CDW structure of the compressible liquid (Fig. 1) makes such a pinning possible even though the disorder is long range. Note that, although the pinning prohibits sliding of the CDW as a whole, the current can still flow along the boundaries of the filled and empty 1 regions (the “bulk edge states”). Precisely at nN ­ 2 , the bulk edge states form a percolating network, which leads to a narrow peak in conductivity with, in certain models , a universal height 0.5e2 yh. We start our analysis by writing down the HF cohesive energy of the electrons at the upper partially filled LL 15 JANUARY 1996 (cf. Refs. [1,2]), nL X 2 e ũHF sqd jDsqdj . (7) 2nN qfi0 Here and below we use tilde for Fourier transformed quantities, L is the size of the system, nL ­ s2pl 2 d21 , and Dsrd is the CDW order parameter. It is proportional to the guiding center density at the point r. For instance, the WC corresponds to Dsrd in the form  ∑ ∏ sr 2 Ri d2 2 X exp 2 . (8) Dsrd ø 2 L i l2 The HF interaction potential ũHF sqd entering Eq. (7) is the difference of the direct and the exchange terms, ũHF sqd ­ ũH sqd 2 ũex sqd, which are further defined by e2 F 2 sqd nLũex sqd ­ uH sql 2 d, nLũH sqd ­ , (9) q´sqdl 2 CDW Ecoh ­ Fsqd ­ e2q l y4 LN sq2 l 2 y2d , (10) LN being the Laguerre polynomial. Following Ref.  (see also Ref. ), the screening by the lower LL’s is explicitly taken into account with the help of the dielectric constant æ Ω 2 2 f1 2 J0 sqRc dg . (11) ´sqd ­ k 1 1 qaB From Eqs. (9) and (10) an asymptotic expression for ũHF sqd can be derived, ∂ µ æ Ω rs21 h̄vc rs sins2qRc d 1 p nL ũHF sqd ø 2 p ln 1 1 p (12) 1 2 Eh . p 2qRc 2 2 qRc 2qRc f1 1 srs y 2 dg We want to find the distribution of the guiding center density Dsx, yd that minimizes the energy. Generally, this is a nontrivial problem because the HF equations have to be solved self-consistently. However, if the CDW is unidirectional, i.e., if Dsx, yd does not depend on y, the self-consistency condition is simply Dsxd ­ Qf2eHF sxdgyL2 , X e x̂deiqx , nLũHF sqdDsq eHF sxd ­ (13) (14) qfi0 where eHF sxd is the HF self-energy, Qsxd is the step function, and x̂ is a unit vector in the x direction. The meaning of this condition is that the states above the Fermi level are empty and below the Fermi level are filled. For N . 0 the Hartree potential ũH sqd necessarily has zeros due to the factor Fsqd containing the Laguerre polynomial [Eqs. (9) and (10)]. The first zero, q0 , is approximately given by q0 ø 2.4yRc . The exchange potential is always positive; hence, there exist q where the total HF potential ũHF is negative [in agreement with Eq. (12)]. This leads to the CDW instability because the energy can be reduced by creating a perturbation at any of such wave vectors (cf. Ref. ). In the parameter range 0.06 , rs , 1 and N , 50 well covering all cases of practical interest, the HF potential is negative at all q . q0 and reaches its lowest value near 2 2 q ­ q0 (see Fig. 2). One can guess then that the lowest energy CDW is the one with the largest possible [under e 0 x̂dj. The CDW the conditions (13) and (14)] value of jDsq having this property consists of alternating strips Dsxd ­ 0 e and Dsxd ­ 1yL2 [Fig. 1(a)], and nonzero Dsqd are given by ∂ µ e x̂d ­ q0 sin pnN q , Dsq (15) pq q0 provided q is an integer multiple of q0 . Our numerical 1 simulations showed that at nN close to 2 this is indeed the correct type of the solution in the specified above range of rs and N, but q0 should be replaced by a slightly FIG. 2. The Hartree, exchange, and HF potentials in q space for N ­ 5 and rs ­ 0.5. 501 VOLUME 76, NUMBER 3 PHYSICAL REVIEW LETTERS smaller value of 2.3yRc corresponding to the spatial period of 2.7Rc . Having established the functional form of Dsxd, let CDW . Performing the us estimate the cohesive energy Ecoh summation in Eq. (7) with the help of Eqs. (12) and (15), one recovers Eq. (5). As for the DOS, it is given by snL q0 ypd jdeHF ydxj21 . It can be verified that eHF sxd reaches its lowest and largest values at x ­ 0 and x ­ pyq0 , respectively. These extrema result in the Van Hove singularities at the edges of the spectrum separated by the gap Eg ­ 2jes0dj. Equation (6) now follows from Eqs. (12), (14), and (15). So far we discussed the unidirectional CDW, which can be analyzed at least partially analytically. 2D CDW patterns were studied numerically. We restricted the choice of Dsrd to the form (8) suggested by the WC state. Recall that in the WC state Ri coincide with the sites of a triangular lattice with density nN nL. In the simulations we used a different set of Ri , corresponding to the triangular lattice with the density nL . The fraction nN of the total of 50 3 50 lattice sites was initially randomly populated, and then the energy was numerically minimized with respect to different rearrangements of the populated sites. The expression for the energy follows from Eqs. (7) and (8): 1 X gHF sRi 2 Rj d sni 2 nN d snj 2 nN d , (16) Eø 2 i,j 1 eHF sqd ­ exps2 2 q2 l 2 d ũHF sqd and ni is the occuwhere g pancy of the ith site. In this notation the energy has a transparent interpretation of pairwise interaction among the y single-electron states jcRi l. 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